The considerable overlap in the dielectric properties of benign and malignant tissue at microwave frequencies means that breast tumour classification using traditional UWB Radar imaging algorithms could be very problematic. Several studies have examined the possibility of using the Radar Target Signature (RTS) of a tumour to classify the tumour as either benign or malignant, since the RTS has been shown to be influenced by the size, shape and surface texture of tumours. The main weakness of existing studies is that they mainly consider tumours in a 3D dielectrically homogenous or 2D heterogeneous breast model. In this paper, the effects of dielectric heterogeneity on a novel Spiking Neural Network (SNN) classifier are examined in terms of both sensitivity and specificity, using a 3D dielectrically heterogeneous breast model. The performance of the SNN classifier is compared to an existing LDA classifier. The effect of combining conflicting classification readings in a multi-antenna system is also considered. Finally and importantly, misclassified tumours are analysed and suggestions for future work are discussed.